A numerical root of an expression in a bracket where it changes sign (Brent's method). Returns a record: x, residual, the final bracket (lower, upper), iterations, evaluations and status.
expr (symbolic) -- the function, e.g. x^2 - 2var (string) -- its variable, unquotedlower (number) -- left end of the bracketupper (number) -- right end of the bracketSolves y' = f(t, y) with fixed-step classical Runge-Kutta (RK4). Returns a record with the final t and y, iterations, evaluations and status.
f (symbolic) -- the right-hand side f(t, y), e.g. y - tt (string) -- the independent variable, unquotedy (string) -- the dependent variable, unquotedt0 (number) -- initial ty0 (number) -- initial yh (number) -- step sizesteps (number) -- number of stepsA root by bisection in a bracket where the function changes sign. Slow but always keeps the root bracketed.
f (symbolic) -- the function, e.g. x^2 - 2var (string) -- its variable, unquotedlower (number) -- left endupper (number) -- right endtol (number) -- optional: tolerance for the result, e.g. 1e-6 (default 1e-12)A root by Newton's method from a starting guess. Needs the derivative; not guaranteed to converge.
f (symbolic) -- the function, e.g. x^2 - 2var (string) -- its variable, unquoteddf (symbolic) -- the derivative, e.g. 2*xdvar (string) -- its variablex0 (number) -- starting guesstol (number) -- optional: tolerance for the result, e.g. 1e-6 (default 1e-12)A root by the secant method from two starting guesses; not guaranteed to converge.
f (symbolic) -- the function, e.g. x^2 - 2var (string) -- its variable, unquotedx0 (number) -- first guessx1 (number) -- second guesstol (number) -- optional: tolerance for the result, e.g. 1e-6 (default 1e-12)A numerical derivative by central differences with Richardson extrapolation, with an error estimate.
f (symbolic) -- the function, e.g. x^2 - 2var (string) -- its variable, unquotedat (number) -- the pointh (number) -- the step, e.g. 0.001tol (number) -- optional: tolerance for the result, e.g. 1e-6 (default 1e-12)A numerical derivative by forward differences with Richardson extrapolation.
f (symbolic) -- the function, e.g. x^2 - 2var (string) -- its variable, unquotedat (number) -- the pointh (number) -- the steptol (number) -- optional: tolerance for the result, e.g. 1e-6 (default 1e-12)A numerical definite integral by adaptive Simpson's rule, with an error estimate.
f (symbolic) -- the function, e.g. x^2 - 2var (string) -- its variable, unquotedlower (number) -- left endupper (number) -- right endtol (number) -- optional: tolerance for the result, e.g. 1e-6 (default 1e-12)The composite trapezoid rule with n panels (refined once to estimate the error).
f (symbolic) -- the function, e.g. x^2 - 2var (string) -- its variable, unquotedlower (number) -- upper (number) -- n (number) -- number of panelstol (number) -- optional: tolerance for the result, e.g. 1e-6 (default 1e-12)The composite Simpson rule with n panels (n even; refined once to estimate the error).
f (symbolic) -- the function, e.g. x^2 - 2var (string) -- its variable, unquotedlower (number) -- upper (number) -- n (number) -- number of panels, eventol (number) -- optional: tolerance for the result, e.g. 1e-6 (default 1e-12)Linear interpolation: the value at x on the line through (x0, y0) and (x1, y1).
x0 (number) -- y0 (number) -- x1 (number) -- y1 (number) -- x (number) -- Solves y' = f(t, y) with fixed-step Euler steps.
f (symbolic) -- the right-hand side f(t, y), e.g. y - tt (string) -- the independent variabley (string) -- the dependent variablet0 (number) -- initial ty0 (number) -- initial yh (number) -- step sizesteps (number) -- number of stepstol (number) -- optional: tolerance for the result, e.g. 1e-6 (default 1e-12)Solves y' = f(t, y) up to a target t with adaptive Runge-Kutta-Fehlberg 4(5) steps.
f (symbolic) -- the right-hand side f(t, y), e.g. y - tt (string) -- the independent variabley (string) -- the dependent variablet0 (number) -- initial ty0 (number) -- initial yt_end (number) -- the target th0 (number) -- first step sizetol (number) -- optional: tolerance for the result, e.g. 1e-6 (default 1e-12)The minimum of a function on an interval by Brent's method (the function should have one minimum there).
f (symbolic) -- the function, e.g. x^2 - 2var (string) -- its variable, unquotedlower (number) -- upper (number) -- tol (number) -- optional: tolerance for the result, e.g. 1e-6 (default 1e-12)The minimum of a function on an interval by golden-section search.
f (symbolic) -- the function, e.g. x^2 - 2var (string) -- its variable, unquotedlower (number) -- upper (number) -- tol (number) -- optional: tolerance for the result, e.g. 1e-6 (default 1e-12)The least-squares line y = intercept + slope*x through data points, with the residual sum of squares.
xs (array) -- x valuesys (array) -- y valuesNewton's method for two equations f1 = 0, f2 = 0 in two variables, from a starting point; the Jacobian is exact (symbolic).
equations (expression list) -- [f1, f2]x (name) -- y (name) -- x0 (number) -- y0 (number) -- The parameter value in [lower, upper] that minimizes an objective (e.g. a sum of squared errors).
objective (expression) -- p (name) -- lower (number) -- upper (number) -- The interpolating polynomial through the points (xs, ys), in Newton's divided-difference form, evaluated at x.
xs (array) -- ys (array) -- x (number) -- The natural cubic spline through the points (xs, ys), evaluated at x.
xs (array) -- ys (array) -- x (number) -- The discrete Fourier transform of a list whose length is a power of 2 (real or complex).
zs (array) -- The inverse discrete Fourier transform.
zs (array) --